Theorem ceqsalv 3500

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 2219. (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 1969	. 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 1846	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))
5		ceqsalv.1	. . . 4 ⊢ 𝐴 ∈ V
6	5	isseti 3479	. . 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)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-clel 2844

This theorem is referenced by:  ceqsexv  3509  ralxpxfr2d  3612  frsn  5750  raliunxp  5826  idrefALT  6114  funimass4  6946  fnssintima  7361  imaeqalov  7650  marypha2lem3  9397  kmlem12  10145  vdwmc2  17039  itg2leub  25862  eqcuts2  27945  addsuniflem  28160  mulsuniflem  28308  onsfi  28515  nmoubi  31065  choc0  31619  nmopub  32201  nmfnleub  32218  elintfv  36190  heibor1lem  38383  elmapintrab  44229