Theorem rexeqbi1dv 3337

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	rexeqbidvv 3334	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-rex 3069

This theorem is referenced by:  rexeqOLD  3339  friOLD  5647  frsn  5776  isofrlem  7360  f1oweALT  7996  frxp  8150  frxp2  8168  1sdomOLD  9283  oieq2  9551  zfregcl  9632  frmin  9787  hashge2el2difr  14517  cat1  18151  ishaus  23346  isreg  23356  isnrm  23359  lebnumlem3  25009  1vwmgr  30305  3vfriswmgr  30307  isgrpo  30526  pjhth  31422  bnj1154  34992  satfvsuc  35346  satf0suc  35361  sat1el2xp  35364  fmlasuc0  35369  isexid2  37842  ismndo2  37861  rngomndo  37922  stoweidlem28  45984  prprval  47439