Theorem ceqex 3612

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 417	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
3		eqvisset 3475	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
4		alexeqg 3611	. . . 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455

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-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457

This theorem is referenced by:  ceqsexg  3613