Theorem ceqsalv 3520

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 2176. (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 1941	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
2		ceqsalv.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	pm5.74i 271	. . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓))
4	3	albii 1818	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))
5		ceqsalv.1	. . . 4 ⊢ 𝐴 ∈ V
6	5	isseti 3497	. . 3 ⊢ ∃𝑥 𝑥 = 𝐴
7	6	a1bi 362	. 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
8	1, 4, 7	3bitr4i 303	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-clel 2815

This theorem is referenced by:  ceqsexv  3531  ralxpxfr2d  3645  frsn  5772  raliunxp  5849  idrefALT  6130  funimass4  6972  fnssintima  7383  imaeqalov  7673  marypha2lem3  9478  kmlem12  10203  vdwmc2  17018  itg2leub  25770  eqscut2  27852  addsuniflem  28035  mulsuniflem  28176  nmoubi  30792  choc0  31346  nmopub  31928  nmfnleub  31945  elintfv  35766  heibor1lem  37817  elmapintrab  43594