Theorem ceqsalv 3492

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 2215. (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 1975	. 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 1852	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))
5		ceqsalv.1	. . . 4 ⊢ 𝐴 ∈ V
6	5	isseti 3471	. . 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 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2837

This theorem is used by:  ceqsexv  3501  ralxpxfr2d  3603  frsn  5747  raliunxp  5823  idrefALT  6111  funimass4  6946  fnssintima  7368  imaeqalov  7656  marypha2lem3  9410  kmlem12  10167  vdwmc2  17075  itg2leub  25963  eqcuts2  28049  addsuniflem  28264  mulsuniflem  28412  onsfi  28619  nmoubi  31239  choc0  31793  nmopub  32375  nmfnleub  32392  elintfv  36331  heibor1lem  38546  elmapintrab  44403