Theorem ceqex 3638

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 2170	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
2	1	ex 412	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
3		eqvisset 3489	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
4		alexeqg 3637	. . . 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 2172	. . . 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 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099  Vcvv 3471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473

This theorem is referenced by:  ceqsexg  3639