Theorem ixpeq2 4416

Description: Equality theorem for infinite Cartesian product.

Assertion

Ref	Expression
ixpeq2	|- (A.x e. A B = C -> X_x e. A B = X_x e. A C)

Proof of Theorem ixpeq2

Step	Hyp	Ref	Expression
1		ss2ixp 4415	. . 3 |- (A.x e. A B (_ C -> X_x e. A B (_ X_x e. A C)
2		ss2ixp 4415	. . 3 |- (A.x e. A C (_ B -> X_x e. A C (_ X_x e. A B)
3	1, 2	anim12i 340	. 2 |- ((A.x e. A B (_ C /\ A.x e. A C (_ B) -> (X_x e. A B (_ X_x e. A C /\ X_x e. A C (_ X_x e. A B))
4		eqss 2128	. . . 4 |- (B = C <-> (B (_ C /\ C (_ B))
5	4	ralbii 1714	. . 3 |- (A.x e. A B = C <-> A.x e. A (B (_ C /\ C (_ B))
6		r19.26 1797	. . 3 |- (A.x e. A (B (_ C /\ C (_ B) <-> (A.x e. A B (_ C /\ A.x e. A C (_ B))
7	5, 6	bitri 180	. 2 |- (A.x e. A B = C <-> (A.x e. A B (_ C /\ A.x e. A C (_ B))
8		eqss 2128	. 2 |- (X_x e. A B = X_x e. A C <-> (X_x e. A B (_ X_x e. A C /\ X_x e. A C (_ X_x e. A B))
9	3, 7, 8	3imtr4i 226	1 |- (A.x e. A B = C -> X_x e. A B = X_x e. A C)

Syntax hints:   -> wi 3   /\ wa 230   = wceq 997  A.wral 1692   (_ wss 2098  X_cixp 4408

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-id 2891  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-fv 3255  df-ixp 4409

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