Theorem ax11i 1140

Description: Inference that has ax-11 969 (without A.y) as its conclusion and doesn't require ax-10 968, ax-11 969, or ax-12 970 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70.

Hypotheses

Ref	Expression
ax11i.1	|- (x = y -> (ph <-> ps))
ax11i.2	|- (ps -> A.xps)

Assertion

Ref	Expression
ax11i	|- (x = y -> (ph -> A.x(x = y -> ph)))

Proof of Theorem ax11i

Step	Hyp	Ref	Expression
1		ax11i.1	. 2 |- (x = y -> (ph <-> ps))
2		ax11i.2	. . 3 |- (ps -> A.xps)
3	1	biimprcd 156	. . 3 |- (ps -> (x = y -> ph))
4	2, 3	19.21ai 1000	. 2 |- (ps -> A.x(x = y -> ph))
5	1, 4	syl6bi 214	1 |- (x = y -> (ph -> A.x(x = y -> ph)))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   = wceq 958

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147

Copyright terms: Public domain