Theorem ceqex 3605

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 2217	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
2	1	ex 418	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
3		eqvisset 3470	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
4		alexeqg 3604	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
5	3, 4	syl 18	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
6		sp 2219	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜑))
7	6	com12 33	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜑))
8	5, 7	sylbird 263	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜑))
9	2, 8	impbid 215	1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452

This theorem is used by:  ceqsexg  3606