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Metamath Proof Explorer

Theorem sbceq1d

Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017) (Revised by NM, 30-Jun-2018)

Ref Expression
Hypothesis sbceq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion sbceq1d ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜓 ) )

Proof

Step Hyp Ref Expression
1 sbceq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 dfsbcq ⊢ ( 𝐴 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜓 ) )
3 1 2 syl ⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜓 ) )