Theorem bnd 4733

Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 3620), derived from the Collection Principle cp 4732. Its strength lies in the rather profound fact that ph(x, y) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom.

Assertion

Ref	Expression
bnd	|- (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph)

Distinct variable groups:   ph,z,w   x,y,z,w

Proof of Theorem bnd

Step	Hyp	Ref	Expression
1		cp 4732	. . 3 |- E.wA.x e. z (E.yph -> E.y e. w ph)
2		r19.20 1705	. . . 4 |- (A.x e. z (E.yph -> E.y e. w ph) -> (A.x e. z E.yph -> A.x e. z E.y e. w ph))
3	2	19.22i 1042	. . 3 |- (E.wA.x e. z (E.yph -> E.y e. w ph) -> E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph))
4	1, 3	ax-mp 7	. 2 |- E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph)
5		19.37v 1305	. 2 |- (E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph) <-> (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph))
6	4, 5	mpbi 189	1 |- (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph)

Syntax hints:   -> wi 3  E.wex 982  A.wral 1648  E.wrex 1649

This theorem is referenced by:  bnd2 4734

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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