Theorem raleq1f 1783

Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions.

Hypotheses

Ref	Expression
raleq1f.1	|- (y e. A -> A.x y e. A)
raleq1f.2	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
raleq1f	|- (A = B -> (A.x e. A ph <-> A.x e. B ph))

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem raleq1f

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 |- (y e. A -> A.x y e. A)
2		raleq1f.2	. . . 4 |- (y e. B -> A.x y e. B)
3	1, 2	hbeq 1565	. . 3 |- (A = B -> A.x A = B)
4		eleq2 1535	. . . 4 |- (A = B -> (x e. A <-> x e. B))
5	4	imbi1d 613	. . 3 |- (A = B -> ((x e. A -> ph) <-> (x e. B -> ph)))
6	3, 5	albid 1104	. 2 |- (A = B -> (A.x(x e. A -> ph) <-> A.x(x e. B -> ph)))
7		df-ral 1649	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
8		df-ral 1649	. 2 |- (A.x e. B ph <-> A.x(x e. B -> ph))
9	6, 7, 8	3bitr4g 555	1 |- (A = B -> (A.x e. A ph <-> A.x e. B ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645

This theorem is referenced by:  raleq1 1786  hta 4728

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472  df-ral 1649

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