Theorem equsexALT 2424

Description: Alternate proof of equsex 2423. This proves the result directly, instead of as a corollary of equsal 2422 via equs4 2421. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2377 is ax6e 2388. This proof mimics that of equsal 2422 (in particular, note that pm5.32i 574, exbii 1848, 19.41 2236, mpbiran 709 correspond respectively to pm5.74i 271, albii 1819, 19.23 2212, a1bi 362). (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 574	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝑦 ∧ 𝜓))
3	2	exbii 1848	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))
4		ax6e 2388	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
5		equsal.1	. . . 4 ⊢ Ⅎ𝑥𝜓
6	5	19.41 2236	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜓))
7	4, 6	mpbiran 709	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜓)
8	3, 7	bitri 275	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)