Theorem ceqsalv 3489

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid ax-12 2212. (Revised by SN, 8-Sep-2024.)

Hypotheses

Ref	Expression
ceqsalv.1	⊢ 𝐴 ∈ V
ceqsalv.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ceqsalv	⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsalv

Step	Hyp	Ref	Expression
1		19.23v 1974	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
2		ceqsalv.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	pm5.74i 274	. . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓))
4	3	albii 1851	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))
5		ceqsalv.1	. . . 4 ⊢ 𝐴 ∈ V
6	5	isseti 3468	. . 3 ⊢ ∃𝑥 𝑥 = 𝐴
7	6	a1bi 365	. 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
8	1, 4, 7	3bitr4i 306	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-clel 2835

This theorem is used by:  ceqsexv  3498  ralxpxfr2d  3599  frsn  5738  raliunxp  5815  idrefALT  6106  funimass4  6949  fnssintima  7372  imaeqalov  7660  marypha2lem3  9429  kmlem12  10240  vdwmc2  17157  itg2leub  26055  eqcuts2  28172  addsuniflem  28387  mulsuniflem  28535  onsfi  28742  nmoubi  31374  choc0  31928  nmopub  32510  nmfnleub  32527  elintfv  36530  heibor1lem  38744  elmapintrab  44577