Theorem mpteq2dv 3869

Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)

Hypothesis

Ref	Expression
mpteq2dv.1	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
mpteq2dv	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem mpteq2dv

Step	Hyp	Ref	Expression
1		mpteq2dv.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
2	1	adantr 270	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	mpteq2dva 3868	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433   ↦ cmpt 3839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-opab 3840  df-mpt 3841

This theorem is referenced by:  ofeq  5734  rdgeq1  5981  rdgeq2  5982  omv  6058  oeiv  6059