Theorem ceqsalv 3479

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

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		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsalv.1	. 2 ⊢ 𝐴 ∈ V
3		ceqsalv.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	ceqsal 3478	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870

This theorem is referenced by:  ralxpxfr2d  3587  clel4  3604  frsn  5603  raliunxp  5674  idrefALT  5940  funimass4  6705  marypha2lem3  8885  kmlem12  9572  vdwmc2  16305  itg2leub  24338  nmoubi  28555  choc0  29109  nmopub  29691  nmfnleub  29708  elintfv  33120  heibor1lem  35247  elmapintrab  40276