Theorem rexeq12d 1842

Description: Equality deduction for restricted universal quantifier.

Hypotheses

Ref	Expression
raleq12d.1	⊢ (φ → A = B)
raleq12d.2	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
rexeq12d	⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))

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

Proof of Theorem rexeq12d

Step	Hyp	Ref	Expression
1		raleq12d.1	. . 3 ⊢ (φ → A = B)
2	1	rexeq1d 1837	. 2 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B ψ))
3		raleq12d.2	. . 3 ⊢ (φ → (ψ ↔ χ))
4	3	rexbidv 1711	. 2 ⊢ (φ → (∃x ∈ B ψ ↔ ∃x ∈ B χ))
5	2, 4	bitrd 539	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))

Syntax hints:   → wi 3   ↔ wb 153   = wceq 997  ∃wrex 1693

This theorem is referenced by:  iscms 8031  isring 8225  ringi 8226  nmofval 8509  isfuna 10838

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-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-cleq 1515  df-clel 1518  df-rex 1697

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