Theorem ceqsalg 3490

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3491. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.)

Hypotheses

Ref	Expression
ceqsalg.1	⊢ Ⅎ𝑥𝜓
ceqsalg.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ceqsalg	⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		ceqsalg.1	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsalg.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	ax-gen 1825	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		ceqsalt 3488	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
5	1, 3, 4	mp3an12 1480	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-clel 2838

This theorem is referenced by:  ceqsalALT  3493  uniiunlem  4042  ralrnmpo  7551  fimaxre3  12162  pmapglbx  40521