Theorem equsexvw 2011

Description: Version of equsexv 2269 with a disjoint variable condition, and of equsex 2440 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2010. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)

Hypothesis

Ref	Expression
equsalvw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem equsexvw

Step	Hyp	Ref	Expression
1		alinexa 1843	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		equsalvw.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	notbid 320	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	equsalvw 2010	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ 𝜓)
5	1, 4	bitr3i 279	. 2 ⊢ (¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ¬ 𝜓)
6	5	con4bii 323	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  equvinv  2036  cleljust  2123  sbelx  2255  cleljustab  2804  sbhypf  3554  axsepgfromrep  5203  dfid3  5464  opeliunxp  5621  imai  5944  coi1  6117  opabex3d  7668  opabex3rd  7669  opabex3  7670  fsplit  7814  fsplitOLD  7815  mapsnend  8590  dfac5lem1  9551  dfac5lem3  9553  dffix2  33368  sscoid  33376  elfuns  33378  pmapglb  36908  polval2N  37044  opeliun2xp  44388