Theorem rexeqdv 2512

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)

Hypothesis

Ref	Expression
raleq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rexeqdv

Step	Hyp	Ref	Expression
1		raleq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		rexeq 2506	. 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
3	1, 2	syl 14	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312

This theorem is referenced by:  rexeqbidv  2518  rexeqbidva  2520  fnunirn  5406  cbvexfo  5426  genipv  6607