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Metamath Proof Explorer

Theorem sbceq1dd

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

Ref Expression
Hypotheses sbceq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
sbceq1dd.2 ⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 )
Assertion sbceq1dd ( 𝜑 → [ 𝐵 / 𝑥 ] 𝜓 )

Proof

Step Hyp Ref Expression
1 sbceq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 sbceq1dd.2 ⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 )
3 1 sbceq1d ⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜓 ) )
4 2 3 mpbid ⊢ ( 𝜑 → [ 𝐵 / 𝑥 ] 𝜓 )