Theorem cbvrexv2 3204

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
cbvralv2.1	⊢ (x = y → (ψ ↔ χ))
cbvralv2.2	⊢ (x = y → A = B)

Assertion

Ref	Expression
cbvrexv2	⊢ (∃x ∈ A ψ ↔ ∃y ∈ B χ)

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

Allowed substitution hints:   ψ(x)   χ(y)   A(x)   B(y)

Proof of Theorem cbvrexv2

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲyA
2		nfcv 2490	. 2 ⊢ ℲxB
3		nfv 1619	. 2 ⊢ Ⅎyψ
4		nfv 1619	. 2 ⊢ Ⅎxχ
5		cbvralv2.2	. 2 ⊢ (x = y → A = B)
6		cbvralv2.1	. 2 ⊢ (x = y → (ψ ↔ χ))
7	1, 2, 3, 4, 5, 6	cbvrexcsf 3200	1 ⊢ (∃x ∈ A ψ ↔ ∃y ∈ B χ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)