Theorem eleq1w

Description: Weaker version of eleq1 (but more general than elequ1 ) not depending on ax-ext nor df-cleq .

Note that this provides a proof of ax-8 from Tarski's FOL and dfclel (simply consider an instance where A is replaced by a setvar and deduce the forward implication by biimpd ), which shows that dfclel is too powerful to be used as a definition instead of df-clel . (Contributed by BJ, 24-Jun-2019)

	Ref	Expression
Assertion	eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		equequ2	⊢ ( 𝑥 = 𝑦 → ( 𝑧 = 𝑥 ↔ 𝑧 = 𝑦 ) )
2	1	anbi1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴 ) ↔ ( 𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴 ) ) )
3	2	exbidv	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑧 ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴 ) ↔ ∃ 𝑧 ( 𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴 ) ) )
4		dfclel	⊢ ( 𝑥 ∈ 𝐴 ↔ ∃ 𝑧 ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝐴 ) )
5		dfclel	⊢ ( 𝑦 ∈ 𝐴 ↔ ∃ 𝑧 ( 𝑧 = 𝑦 ∧ 𝑧 ∈ 𝐴 ) )
6	3 4 5	3bitr4g	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )