Theorem ceqex 3633

Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.)

Assertion

Ref	Expression
ceqex	⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqex

Step	Hyp	Ref	Expression
1		19.8a 2166	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
2	1	ex 412	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
3		eqvisset 3484	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
4		alexeqg 3632	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
5	3, 4	syl 17	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
6		sp 2168	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜑))
7	6	com12 32	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜑))
8	5, 7	sylbird 260	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜑))
9	2, 8	impbid 211	1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468

This theorem is referenced by:  ceqsexg  3634