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Metamath Proof Explorer

Theorem equsexALT

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

Ref Expression
Hypotheses equsal.1 ⊢ Ⅎ 𝑥 𝜓
equsal.2 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion equsexALT ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )

Proof

Step Hyp Ref Expression
1 equsal.1 ⊢ Ⅎ 𝑥 𝜓
2 equsal.2 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3 2 pm5.32i ⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ( 𝑥 = 𝑦 ∧ 𝜓 ) )
4 3 exbii ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) )
5 ax6e ⊢ ∃ 𝑥 𝑥 = 𝑦
6 1 19.41 ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝑦 ∧ 𝜓 ) )
7 5 6 mpbiran ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) ↔ 𝜓 )
8 4 7 bitri ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )