Theorem rexeqbi1dv 3357

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) (Proof shortened by Steven Nguyen, 5-May-2023.)

Hypothesis

Ref	Expression
raleqbi1dv.1	⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexeqbi1dv	⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem rexeqbi1dv

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
2		raleqbi1dv.1	. 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	1, 2	rexeqbidv 3355	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-rex 3112

This theorem is referenced by:  rexeq  3359  fri  5481  frsn  5603  isofrlem  7072  f1oweALT  7655  frxp  7803  1sdom  8705  oieq2  8961  zfregcl  9042  hashge2el2difr  13835  ishaus  21927  isreg  21937  isnrm  21940  lebnumlem3  23568  1vwmgr  28061  3vfriswmgr  28063  isgrpo  28280  pjhth  29176  bnj1154  32381  satfvsuc  32721  satf0suc  32736  sat1el2xp  32739  fmlasuc0  32744  frmin  33197  isexid2  35293  ismndo2  35312  rngomndo  35373  stoweidlem28  42670  prprval  44031