Theorem rexeqbi1dv 3330

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)

Hypothesis

Ref	Expression
raleqd.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		rexeq 3322	. 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
2		raleqd.1	. . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	rexbidv 3233	. 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓))
4	1, 3	bitrd 271	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653  ∃wrex 3090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095

This theorem is referenced by:  fri  5274  frsn  5394  isofrlem  6818  f1oweALT  7385  frxp  7524  1sdom  8405  oieq2  8660  zfregcl  8741  ishaus  21455  isreg  21465  isnrm  21468  lebnumlem3  23090  1vwmgr  27625  3vfriswmgr  27627  isgrpo  27877  pjhth  28777  bnj1154  31584  frmin  32255  isexid2  34141  ismndo2  34160  rngomndo  34221  stoweidlem28  40988