Theorem rexeqbi1dv 3308

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 3306	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  ∃wrex 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-rex 3064

This theorem is referenced by:  frsn  5706  isofrlem  7284  f1oweALT  7914  frxp  8066  frxp2  8084  oieq2  9418  zfregcl  9499  zfregclOLD  9500  frmin  9664  hashge2el2difr  14434  cat1  18055  ishaus  23305  isreg  23315  isnrm  23318  lebnumlem3  24948  1vwmgr  30364  3vfriswmgr  30366  isgrpo  30586  pjhth  31482  bnj1154  35181  satfvsuc  35589  satf0suc  35604  sat1el2xp  35607  fmlasuc0  35612  isexid2  38222  ismndo2  38241  rngomndo  38302  relpfrlem  45397  stoweidlem28  46471  prprval  47989