Theorem equsexALT 2451

Description: Alternate proof of equsex 2450. This proves the result directly, instead of as a corollary of equsal 2449 via equs4 2448. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2404 is ax6e 2415. This proof mimics that of equsal 2449 (in particular, note that pm5.32i 582, exbii 1869, 19.41 2271, mpbiran 719 correspond respectively to pm5.74i 273, albii 1840, 19.23 2247, a1bi 364). (Contributed by BJ, 20-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem equsexALT

Step	Hyp	Ref	Expression
1		equsal.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	pm5.32i 582	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝑦 ∧ 𝜓))
3	2	exbii 1869	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))
4		ax6e 2415	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
5		equsal.1	. . . 4 ⊢ Ⅎ𝑥𝜓
6	5	19.41 2271	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜓))
7	4, 6	mpbiran 719	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜓)
8	3, 7	bitri 277	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∃wex 1800  Ⅎwnf 1804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-nf 1805

This theorem is referenced by: (None)