Theorem equsex 1962

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
equsex.1	⊢ Ⅎxψ
equsex.2	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
equsex	⊢ (∃x(x = y ∧ φ) ↔ ψ)

Proof of Theorem equsex

Step	Hyp	Ref	Expression
1		exnal 1574	. 2 ⊢ (∃x ¬ (x = y → ¬ φ) ↔ ¬ ∀x(x = y → ¬ φ))
2		df-an 360	. . 3 ⊢ ((x = y ∧ φ) ↔ ¬ (x = y → ¬ φ))
3	2	exbii 1582	. 2 ⊢ (∃x(x = y ∧ φ) ↔ ∃x ¬ (x = y → ¬ φ))
4		equsex.1	. . . . 5 ⊢ Ⅎxψ
5	4	nfn 1793	. . . 4 ⊢ Ⅎx ¬ ψ
6		equsex.2	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
7	6	notbid 285	. . . 4 ⊢ (x = y → (¬ φ ↔ ¬ ψ))
8	5, 7	equsal 1960	. . 3 ⊢ (∀x(x = y → ¬ φ) ↔ ¬ ψ)
9	8	con2bii 322	. 2 ⊢ (ψ ↔ ¬ ∀x(x = y → ¬ φ))
10	1, 3, 9	3bitr4i 268	1 ⊢ (∃x(x = y ∧ φ) ↔ ψ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  equsexh  1963  cleljustALT  2015  sb56  2098  sb10f  2122